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Properties

Label 1-211-211.188-r0-0-0
Degree $1$
Conductor $211$
Sign $0.395 + 0.918i$
Analytic cond. $0.979879$
Root an. cond. $0.979879$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (−0.809 + 0.587i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (0.309 + 0.951i)11-s + 12-s + (−0.809 + 0.587i)13-s + 14-s + 15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (−0.809 + 0.587i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.309 − 0.951i)10-s + (0.309 + 0.951i)11-s + 12-s + (−0.809 + 0.587i)13-s + 14-s + 15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(211\)
Sign: $0.395 + 0.918i$
Analytic conductor: \(0.979879\)
Root analytic conductor: \(0.979879\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{211} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 211,\ (0:\ ),\ 0.395 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4264887419 + 0.2806355980i\)
\(L(\frac12)\) \(\approx\) \(0.4264887419 + 0.2806355980i\)
\(L(1)\) \(\approx\) \(0.6288498217 - 0.03501243616i\)
\(L(1)\) \(\approx\) \(0.6288498217 - 0.03501243616i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad211 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.50291647515260312949249770960, −25.769206362682949241556733832705, −24.80348276448163620008364827666, −24.05981064407639234504105182166, −22.76063425935208720334637797985, −21.785449376099632823391830086278, −20.54615168247571397922841034512, −19.85389219334970884081898208710, −19.19470389011132410729830230506, −17.4269044844382292099777511588, −16.95506211386490235390520721596, −16.03541728111825118680661088814, −15.41885690434680246419367052415, −14.06167608081123855978707322783, −13.2763008479489614930643155181, −11.52840791832678946261325533647, −10.35803535383726054101757509015, −9.56929908212910855066375653158, −8.84671360624965325036417617059, −7.83815343711998549215353199401, −6.34510529389829430636784245822, −5.29793691319650340344102877533, −4.15594352154212673784735611012, −2.487590997442412879427572610395, −0.442353829597310308232192541738, 2.00985363206506846129645835708, 2.46485263512555952771855347452, 3.87359263621811540012463513976, 6.321692438046446157030338786278, 6.850328269470631263814678702763, 7.98319754298170161004010381804, 9.210569777631527867134165950295, 9.969735580917298185307693627981, 11.23750139853512209333071122610, 12.34605090423250195666462513702, 12.88134594747542299163222338834, 14.29418446537227127993042271971, 15.23645640834223738666505790018, 16.744656218212704939591741042016, 17.7425198027724378536868698158, 18.37591704281141338813658664259, 19.370316574602253505645520429649, 19.69126172623002272324023638307, 21.09558200850359950333726331552, 22.186296357227188965372790632916, 22.830349812989416988976240605149, 24.394249724732115211374331185287, 25.305539141355502655767713110113, 25.98227431654940718028931441492, 26.56209660417097959191961256345

Graph of the $Z$-function along the critical line